CLHS: Issue REAL-NUMBER-TYPE Writeup

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## Issue REAL-NUMBER-TYPE Writeup

    Issue:        REAL-NUMBER-TYPE
    Forum:	      CLEANUP
    References:   Table 4-1.
    Category:     ADDITION
    Edit history: 04-JAN-89, Version 1 by Bob Cassels, Don Sakahara, Kent Pitman,
                             and John Aspinall
                  08-JAN-89, Version 2 by Bob Cassels -- incorporate
                             Masinter's suggestion and make REAL a CLOS class
                  13-JAN-89, Version 3 by Cassels and Aspinall -- incorporate Marc LeBrun's
                             suggestions clarifying the relationship between CL
                             numeric type names and mathematical names
    	      05-APR-89, Version 4 by Pitman (changes per X3J13)
    Status:	      Accepted v3 Mar-89 by X3J13 (on a 12-3 vote) with
    	      amendments. The proposal as amended is v4.

    Problem Description:

      There is no standard type specifier symbol for the CL type
      '(OR RATIONAL FLOAT).

    Proposal (REAL-NUMBER-TYPE:X3J13-MAR-89):

      Make REAL be a CL data type:

      p.13 "Numbers"

        Add:     The NUMBER data type encompasses all of these kinds of
                 numbers.  For convenience, there are names for some
                 subclasses of numbers.  @i[Integers] and @i[ratios] are of
                 type RATIONAL.  @i[Rational numbers] and @[floating-point
                 numbers] are of type REAL.  @i[Real numbers] and @i[complex
                 numbers] are of type NUMBER.

    	     Although the names of these types were chosen with the
    	     terminology of mathematics in mind, the correspondences
    	     are not always exact.  Integers and ratios model the
    	     corresponding mathematical concepts directly.  Numbers
    	     of the FLOAT type may be used to approximate real
    	     numbers, both rational and irrational.  The REAL type
    	     includes all Common Lisp numbers which represent
    	     mathematical real numbers, though there are
    	     mathematical real numbers (irrational numbers)
    	     which do not have an exact Common Lisp representation.
    	     Only REAL numbers may be ordered using the <, >, <=,
    	     and >= functions.

                 Compatibility note:  The Fortran standard defines the term
                 "real datum" to mean "a processor approximation to the value
                 of a real number."  In practice the Fortran "basic real" type
                 is the floating-point data type Common Lisp calls
                 SINGLE-FLOAT.  The Fortran "double precision" type is
                 Common Lisp's DOUBLE-FLOAT.  The Pascal "real" data type is
                 an "implementation-defined subset of the real numbers."  In
                 practice this is usually a floating-point type, often what
                 Common Lisp calls DOUBLE-FLOAT.

                 A translation of an algorithm written in Fortran or Pascal
                 which uses "real" data usually will use some appropriate
                 precision of Common Lisp's FLOAT type.  Some algorithms may
                 gain accuracy and/or flexibility by using Common Lisp's
                 RATIONAL or REAL types instead.

      p.33 "Overlap, Inclusion, and Disjointness of Types":

        Remove:  The types RATIONAL, FLOAT, and COMPLEX are pairwise
                 disjoint subtypes of NUMBER.

                 Rationale: It might be thought that INTEGER and RATIO ...

                 Rationale: It might be thought that FIXNUM and BIGNUM ...

        Add:     The types RATIONAL and FLOAT are pairwise disjoint subtypes
                 of REAL.

                 The types REAL and COMPLEX are pairwise disjoint subtypes
                 of NUMBER.

                 Rationale: It might be thought that FIXNUM and BIGNUM should
                 form an exhaustive partition of the type INTEGER, that INTEGER
                 and RATIO should form an exhaustive partition of RATIONAL,
                 that RATIONAL and FLOAT should form an exhaustive partition of
                 REAL, and that REAL and COMPLEX should form an exhaustive
                 partition of NUMBER.  These are all purposely avoided in order
                 to permit compatible experimentation with extensions to the
                 Common Lisp number system, such as the idea of adding explicit
                 representations of infinity or of positive and negative infinity.

       p.43 Table 4-1 "Standard Type Specifier Symbols"

        Add:     REAL

       p.49 "Type Specifiers that Abbreviate"

         Add:    (REAL low high)
                 Denotes the set of real numbers between low and high.  ...
                 [As with RATIONAL and FLOAT.]

      Make REAL a built-in CLOS class.

      Add a specific data type predicate REALP which tests for membership in
      this type.  [By analogy with NUMBERP.]

    Test Case:

      If a programmer wishes to test for "a number between 1 and 10", the
      only current CL types would be '(or (rational 1 10) (float 1 10)) or
      something like '(and numberp (not complexp) (satisfies range-1-10))
      with (defun range-1-10 (real) (<= 1 real 10)).  Both of these are
      likely less efficient, and certainly less expressive than '(real 1 10).

    Rationale:

      Mathematics has a name for (OR RATIONAL FLOAT) -- it is "real".
      This class is important because it is all the numbers which can be
      ordered.

      Throughout the "Numbers" chapter, the phrase "non-complex number" is
      used.
      MAX, MIN, p. 198 "The arguments may be any non-complex numbers."
      CIS p. 207 "The argument ... may be any non-complex number."

    Current Practice:

      Probably nobody does this.

    Cost to Implementors:

      Some work is necessary to add this name.  But since the underlying
      type already exists the amount of work should be minimal.

    Cost to Users:

      Since this is an upward-compatible extension, it may be ignored by
      users.

    Cost of Non-Adoption:

      Occasional inconvenience and/or inefficiency.

    Benefits:

      Mathematical clarity.

      Ability to do CLOS method dispatch on the type.

    Aesthetics:

      As mentioned under "rationale," this would be a more concise way to
      express a common programming idiom.

    Discussion:

      The name "non-complex number" is incorrect because future
      implementations may wish to include numerical types which are neither
      complex nor real.  [e.g. pure imaginary numbers or quaternions]

      The name "scalar" is incorrect because the mathematical concept of
      scalar may indeed include complex numbers.

      Fortran and Pascal use the name "real" to mean what CL calls
      SINGLE-FLOAT.  That should cause no significant problem, since a Lisp
      program written using the type REAL will do mathematically what the
      equivalent Fortran program would do.  This is because Fortran's "real"
      data-type is a subtype of the CL REAL type.  The only differences
      might be that the Lisp program could be less efficient and/or more
      accurate.

      A survey of several Fortran and Pascal books shows that the distinction
      between INTEGER and REAL is that REAL numbers may have fractional
      parts, while INTEGERs do not.  Later discussions explain that REALs
      cover a greater range.  Much later discussions cover precision
      considerations, over/underflow, etc.  So the average Fortran or Pascal
      programmer should be completely comfortable with the proposed Lisp
      concept of REAL.

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